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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/12
MATHEMATICS
Paper 1 Pure Mathematics 1 (P1)
May/June 2012
1 hour 45 minutes
*9656618107*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 4 printed pages.
JC12 06_9709_12/RP
© UCLES 2012
[Turn over
2
1
y
y=
O
2
6
2x – 3
3
x
6
, the x-axis and the lines x = 2 and
2x − 3
x = 3. Find, in terms of π , the volume obtained when this region is rotated through 360◦ about the
x-axis.
[4]
The diagram shows the region enclosed by the curve y =
2
√
2
The equation of a curve is y = 4 x + √ .
x
(i) Obtain an expression for
dy
.
dx
[3]
(ii) A point is moving along the curve in such a way that the x-coordinate is increasing at a constant
rate of 0.12 units per second. Find the rate of change of the y-coordinate when x = 4.
[2]
3
The coefficient of x3 in the expansion of (a + x)5 + (2 − x)6 is 90. Find the value of the positive
constant a.
[5]
4
The point A has coordinates (−1, −5) and the point B has coordinates (7, 1). The perpendicular
bisector of AB meets the x-axis at C and the y-axis at D. Calculate the length of CD.
[6]
5
(i) Prove the identity tan x +
(ii) Solve the equation
© UCLES 2012
1
1
≡
.
tan x sin x cos x
2
= 1 + 3 tan x, for 0◦ ≤ x ≤ 180◦ .
sin x cos x
9709/12/M/J/12
[2]
[4]
3
6
A
B
2.4 rad
8 cm
O
The diagram shows a metal plate made by removing a segment from a circle with centre O and radius
8 cm. The line AB is a chord of the circle and angle AOB = 2.4 radians. Find
7
(i) the length of AB,
[2]
(ii) the perimeter of the plate,
[3]
(iii) the area of the plate.
[3]
(a) In an arithmetic progression, the sum of the first n terms, denoted by Sn , is given by
Sn = n2 + 8n.
Find the first term and the common difference.
[3]
(b) In a geometric progression, the second term is 9 less than the first term. The sum of the second
and third terms is 30. Given that all the terms of the progression are positive, find the first term.
[5]
8
(i) Find the angle between the vectors 3i − 4k and 2i + 3j − 6k.
[4]
−−→
The vector OA has a magnitude of 15 units and is in the same direction as the vector 3i − 4k. The
−−→
vector OB has a magnitude of 14 units and is in the same direction as the vector 2i + 3j − 6k.
−−→
−−→
(ii) Express OA and OB in terms of i, j and k.
[3]
−−→
(iii) Find the unit vector in the direction of AB.
[3]
[Questions 9 and 10 are printed on the next page.]
© UCLES 2012
9709/12/M/J/12
[Turn over
4
9
y
A
y = –x2 + 8x –10
B
x
O
The diagram shows part of the curve y = −x2 + 8x − 10 which passes through the points A and B. The
curve has a maximum point at A and the gradient of the line BA is 2.
10
(i) Find the coordinates of A and B.
[7]
(ii) Find ã y dx and hence evaluate the area of the shaded region.
[4]
Functions f and g are defined by
f : x → 2x + 5 for x ∈ >,
8
g : x →
for x ∈ >, x ≠ 3.
x−3
(i) Obtain expressions, in terms of x, for f −1 (x) and g−1 (x), stating the value of x for which g−1 (x)
is not defined.
[4]
(ii) Sketch the graphs of y = f (x) and y = f −1 (x) on the same diagram, making clear the relationship
between the two graphs.
[3]
(iii) Given that the equation fg(x) = 5 − kx, where k is a constant, has no solutions, find the set of
possible values of k.
[5]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of
Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2012
9709/12/M/J/12